Composition laws of binary quadratic forms and isolations of quadratic forms
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918122109272064 |
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| author | Ju, Jangwon Kim, Daejun Kim, Kyoungmin Kim, Mingyu Oh, Byeong-Kweon |
| author_facet | Ju, Jangwon Kim, Daejun Kim, Kyoungmin Kim, Mingyu Oh, Byeong-Kweon |
| contents | A positive definite and integral quadratic form $f$ is called irrecoverable if there is a quadratic form $F$ such that it represents all proper subforms of $f$, whereas it does not represent $f$ itself. In this case, $F$ is called an isolation of $f$. In this article, we prove that there does not exist a binary isolation of any unary quadratic form. We also prove that there does not exist a ternary isolation of any binary quadratic form. Furthermore, if the form class group of a primitive binary quadratic form has no element of order $4$, then the discriminant of any quaternary isolation of it, if exists, is a square of an integer. The composition laws of primitive binary quadratic forms play an essential role in the proofs of the results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07916 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Composition laws of binary quadratic forms and isolations of quadratic forms Ju, Jangwon Kim, Daejun Kim, Kyoungmin Kim, Mingyu Oh, Byeong-Kweon Number Theory A positive definite and integral quadratic form $f$ is called irrecoverable if there is a quadratic form $F$ such that it represents all proper subforms of $f$, whereas it does not represent $f$ itself. In this case, $F$ is called an isolation of $f$. In this article, we prove that there does not exist a binary isolation of any unary quadratic form. We also prove that there does not exist a ternary isolation of any binary quadratic form. Furthermore, if the form class group of a primitive binary quadratic form has no element of order $4$, then the discriminant of any quaternary isolation of it, if exists, is a square of an integer. The composition laws of primitive binary quadratic forms play an essential role in the proofs of the results. |
| title | Composition laws of binary quadratic forms and isolations of quadratic forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.07916 |